Double copy structure of parity-violating CFT correlators

We show that general parity-violating 3d conformal field theories show a double copy structure for momentum space 3-point functions of conserved currents, stress tensor and marginal scalar operators. Splitting up the CFT correlator into two parts - called homogeneous and non-homogeneous - we show that double copy relations exist for each part separately. We arrive at similar conclusions regarding double copy structures using tree-level correlators of massless fields in $dS_4$. We also discuss the flat space limit of these correlators. We further extend the double copy analysis to correlators involving higher-spin conserved currents, which suggests that the spin-$s$ current correlator can be thought of as $s$ copies of the spin one current correlator.

28 D Momentum space expression of higher spin correlators 28 E Double copy relations in spinor-helicity notation 29 1 Introduction There has been a remarkable confluence in the study of CFT correlators and scattering amplitudes in recent years. As is well known, scattering amplitudes can be extracted by taking a suitable limit of appropriate CFT correlators -in position, momentum or Mellin space [1][2][3][4][5][6]. This enables a CFT derivation of various flat space amplitude results [6,7]. Conversely, amplitude methods have recently been used in the study of CFTs [8,9]. One of the interesting relationships that exists for flat space scattering amplitudes is the double-copy relation between gauge theory and gravity amplitudes, and the associated color-kinematics duality [10][11][12]. Here, substitution of color factors by kinematic factors in the numerators generates gravity amplitudes from gauge amplitudes, thereby manifesting a quadratic relationship between these two theories. This means that amplitudes involving gravitons can be built out from those involving gluons. The double copy relation was first observed in Einstein gravity and pure Yang-Mills theory, and later it was extended to a whole host of theories including higher derivative conformal gravity, higher derivative gauge theories and bi-adjoint scalar theories [13][14][15]. The 3-point structure for the higherderivative theories is significant because they occur in the momentum space form of CFT correlators of conserved currents, stress tensors and scalars. Double copy relations also exist for higher point tree amplitudes and loop amplitudes [16,17]. For a comprehensive review see [18]. The analyses in these works were for the parity-even sector. We will show in this paper that similar relationships between amplitudes continue to hold with the inclusion of possible parity-violating terms in the Lagrangian.
In this work we study 3-point CFT correlators in momentum space. Some recent works where momentum space CFTs have been studied include . In particular, the doublecopy structures of certain parity-even 3-point functions were inferred in momentum space in [33,43]. In three dimensions, in addition to the parity-even structures, one also needs to consider parity-odd correlators. The most general form of 3-point functions in 3d CFTs is known to be of the form : where J s is a spin s conserved current. In position space one can show that the parityeven part can be obtained by adding contributions arising from the free-boson and the free-fermion theories [62][63][64]. However, interacting 3d CFTs such as Chern-Simons-Matter theories can contain a non-trivial parity-odd sector as well [65][66][67][68][69][70]. In this paper we will demonstrate double copy relations between general parity-violating CF T 3 3-point correlators involving marginal scalars, spin one and spin two conserved currents. We will also show that a double-copy like structure exists for correlators involving higher spin conserved currents.
To establish our claim, it is convenient to split up CFT correlators into two parts, namely homogeneous and non-homogeneous parts. Their definition will be made clear in the next section. In particular, we show that under double copy relations, the homogeneous part maps to homogeneous part and the non-homogeneous part maps to non-homogeneous part. Let us illustrate this point by considering T T T , the 3-point function of the stress tensor and JJJ the 3-point function of the conserved spin-1 current. The correlators can be written as : JJJ = JJJ homogeneous + JJJ non-homogeneous

T T T = T T T homogeneous + T T T non-homogeneous
(1. 2) The double copy relation is then given by : where the proportionality factor is momentum dependent and is different for the two cases. It is given explicitly in Section 4.3. We will show that double copy relations hold even with the inclusion of the parity violating contributions. We demonstrate this using the results for parity even CFT correlators from [20,27,29,33] and parity odd correlators from [61,71] where CFT correlators were obtained by solving conformal Ward identities. There is another interesting way to fix the form of these correlators, initiated in [56]. Here, late-time tree level boundary correlators in Lorentzian dS 4 can be computed by first doing an equivalent calculation in flat Minkowski space. Thereafter, using certain conformal properties of the fields in dS 4 , the corresponding dS 4 correlators are obtained by simply dressing the result with a conformal-time integral factor. This Lorentzian dS 4 correlator also naturally computes a Euclidean CF T 3 correlator. We use this method to independently derive the parity-odd structures for the CFT correlators. This method provides a route to obtaining general parity-violating momentum space CF T 3 correlators without solving conformal Ward identities [61,71].
The rest of the paper is organised as follows. In Section 2, we introduce the notation used in this paper. In Section 3, we give the form of all the relevant CFT correlators. In section 4 we study double copy relations between various pairs of correlators. In Section 5, we discuss the flat space limit and write down CFT correlators in terms of tree-level amplitudes without energy conservation for general parity-violating theories of gravitons, gluons and massless scalars in four dimensions. We also discuss here double copy relations for tree level dS 4 correlators and flat space scattering amplitudes. In Section 6 we conclude and give some directions for future study. In Appendix A we give our results for the scattering amplitudes and the CFT correlators in terms of the spinor helicity variables. We give details of a few correlators such as JJT in Appendix B. In Appendix C we give the calculational details used in establishing double copy relations. In Appendix D we give explicit momentum space results for a few correlators involving higher spin currents.

Notation and conventions
In this paper we denote 4-dimensional Lorentzian momenta and polarisation vectors by k µ i and z µ i respectively. Here i is a particle index and µ = 0, 1, 2, 3 is the Lorentz index. For massless spin 2 particles the polarisation tensor can be written as an outer product z µν i = z µ i z ν i . We choose the following gauge to work with null momenta : The 3-dimensional CFT will be Euclidean and current conservation constraints translate to transversality: k i · z i = 0. We will also take z i · z i = 0 which in Euclidean signature implies that the components of z i will be complex.
In our computation we will find it useful to introduce the following notation for various combinations of magnitudes of momenta : We also introduce the following notation : We will make use of spinor-helicity notation. The momentum vector p µ for massless scattering in 4-dimensional flat space-time can be written as p µ σ µ αα = p αα = λ α λα where λ denotes a spinor-helicity variable. Since 4d amplitudes are related to 3d CFT correlators it will be useful to have a 3d version of this formalism by utilising the time-like vector τ µ = (1, 0, 0, 0), or τ αβ = αβ which can be used to go from dotted to undotted indices (see appendix B of [33] and [43]). We use this to defineλ α ≡ τ αβ λβ.
For a correlator comprising conserved currents J s i the conformal Ward identity in spinor-helicity notation takes the following form : where we see that the R.H.S of the equation is given by the transverse Ward identities.
The correlator J s 1 J s 2 J s 3 is given by the sum of two terms that satisfy the homogeneous and non homogeneous equations respectively : The transverse Ward identity is determined in terms of two-point functions. Hence in the momentum space expression for the correlator J s 1 J s 2 J s 3 , we identify the part proportional to the 2-point function coefficient to be the solution to the non-homogeneous Ward identity and the part obtained by setting the coefficient of the 2-point function to zero to be the solution to the homogeneous Ward identity. We will use subscripts h and nh to denote the solutions to the homogeneous and non-homogeneous equations respectively. For a correlator with at least one scalar operator O ∆ with conformal dimension ∆ the conformal Ward identity has a trivial R.H.S and is given by : which holds true both when s 2 = s 3 and s 2 = s 3 . We will denote flat space amplitudes by A. The corresponding correlators in dS 4 1 , with all insertions at the equal-time spatial-slice η = 0, will be denoted by M.

CFT correlators
In this section we present the momentum space expressions for 3-point CF T 3 correlators comprising spin-1 conserved current J, stress tensor T , higher spin conserved currents J s with spin s > 2 and marginal scalar operators O 3 . The parity-even sector of 3-point CFT correlators has been studied by solving the associated conformal Ward identities in a series of works [20,25,27,29]. In [61,71] we studied the parity-odd sector of 3-point correlators by solving conformal Ward identities and using the technique of spin-raising and weightshifting operators in momentum space [58,59,72]. We present our results after contracting the momentum space expressions with null transverse polarization vectors.

3.1
J s J s O 3 for general spin s current In this subsection we write down correlators of the form J s J s O 3 for a general spin s. For s = 1, 2 we can write down their explicit form in momentum space easily. A similar momentum space expression for a correlator involving general spin s current is very cumbersome. However, it takes a very simple form when written in terms of spinor-helicity variables. We also note that, as discussed in the previous section, a correlator of the form J s J s O 3 only has a homogeneous part. The homogeneous correlator is given by the contribution from the parity-even and the parity-odd sectors : Let us first consider the 3-point correlator comprising two spin-1 conserved currents and a marginal scalar operator. The momentum space expression for the parity-even part of the correlation function is [20,29,33] : The momentum space expression for the parity-odd part of the correlator is [61,71] :

T T O 3
Let us now consider the 3-point correlator comprising two stress-tensor insertions and a marginal scalar. The parity-even part of the correlator is given by [20,29,33] : whereas the parity-odd part of this correlator is [71] : Correlators of the form J s J s O 3 have unwieldy expressions in momentum space. The easiest way to determine them is to use weight-shifting operators [59]. For this, we first derive the momentum space expressions for correlators with spin-3 and spin-4 currents using weight-shifting operators [59] : where P (s) i are spin-s projectors [59] and H 12 is a bi-local weight shifting operator which raises the spin at points 1 and 2 and lowers the dimensions at points 1 and 2. In momentum space the operator takes the form [59] : The explicit form of the correlator in (3.6) is complicated and not reproduced here. A similar expression can be written down for parity-odd contribution.
These correlators when expressed in spinor-helicity variables take a very simple form. For this, let us consider the 3-point correlator of two higher spin conserved currents J s with spin s and a marginal scalar. The parity-even part of the correlator is given by [71] : whereas the parity-odd part of the correlator is [71] : As stated earlier, the non-homogeneous piece vanishes for both parity-even and parity-odd correlators : One can check that for the cases when s = 1 and s = 2, the momentum space results in (3.2), (3.3), (3.4), and (3.5) when re-expressed in spinor-helicity variables match the above results.
In this subsection we write down correlators of the form J s J s J s for a general s. Unlike J s J s O , these correlators have both homogeneous and non-homogeneous pieces : We give the explicit forms for JJJ and T T T in momentum space. In both cases, there are exactly two homogeneous pieces, one parity-even and another parity-odd. We also note that there is only one parity-even non-homogeneous contribution. The nonhomogeneous part of the parity-odd correlator is always a contact term. For a general spin s, the momentum space expression is very complicated. However, in spinor-helicity variables it becomes simple and we express the homogeneous part in these variables. The analogous expression for non-homogeneous part is not yet known.

JJJ
Let us now consider the 3-point correlator comprising three spin-1 conserved current insertions. The parity-even part of the correlator is given by [20,27,33] : and by working in a suitable basis we can get rid of the unphysical pole. For instance, it can be shown that after a clever use of 3d degeneracies, one can express the correlator as follows 2 For details of this computation see Appendix C.1.
The parity-odd contribution to the correlator is given by [71] where the homogeneous piece in the form factor is given by : Just as in the parity-even case we see that the form factors have an unphysical extra pole at J = 0. This can again be gotten rid of by working in a suitable basis where it takes the form 3 The non-homogeneous part of the parity-odd correlator is determined by the form factors in (3.18) where c odd T T appears in the parity-odd part of the two point function T T . It can be shown that, the parity-odd non-homogeneous contribution to the correlator is a contact term, see [71] for details. Since the non-homogeneous part is a contact term, this will be ignored while checking double copy relations.

3.2.3
Let us now consider the 3-point correlator comprising three higher spin conserved currents J s with spin s. The parity-even contribution to the correlator is given by [71] : The parity-odd contribution to the correlator is given 4 by [71]

Double copy structure of CFT correlators
In this section we discuss the double copy structure of CFT 3-point correlation functions in momentum space. We will see that the double copy relations are such that homogeneous terms are mapped to homogeneous terms and non-homogeneous terms are mapped to nonhomogeneous terms. We will establish our claims in momentum space for T T O 3 and JJO 3 and for T T T and JJJ . We use the spinor-helicity variables to show this for correlators such as J s J s O 3 and J s J s J s .

T T O 3 and JJO 3
The following double copy structure of T T O 3 even was established in [33] : From the explicit expressions for the correlators in (3.2), (3.3) and (3.5) we notice that the double copy relations extends to the parity-odd sector : Remarkably we also notice from (3.3) and (3.4) that T T O 3 even is also given by the square of JJO 3 odd Let us note that the parity-odd correlator, when written in spinor-helicity variables, gets an extra factor of i as compared to parity-even correlator. This arises due to the fact that when converting momentum space parity-odd answer to spinor-helicity variables, the Levi-Civita tensor ijk is expressed in terms of Pauli matrices and their algebra gives rise to this extra factor. More precisely The above double copy relations for T T O 3 even and T T O 3 odd immediately imply the following double copy structure for the complete correlator : In writing the above double copy relation it is crucial that we have the following relation between JJO 3 even and JJO 3 odd : where c is some constant. As noted in Section 2, for correlators such as T T O 3 and JJO 3 the conformal Ward identity (in spinor helicity variables) does not have a non-homogeneous term. Hence the double copy structure that we obtained above is purely for correlators that satisfy the homogeneous conformal Ward identity.

Double copy for higher-spin correlators
We will now extend our analysis of the double copy structure of T T O 3 to higher spin correlators of the form J s J s O 3 . Using (3.6) one can show that the higher spin correlators take the following form : Calculating the parity-odd contribution to these three point functions is difficult due to the high amount of degeneracy [61,71]. However, in spinor helicity variables the computation becomes easier. In these variables one has the following remarkable relation between the parity-even and parity-odd contributions [71] : for any s and all the other spinor helicity components are zero. Using this one can generalise (4.6) to include the parity-odd sector. The double copy relation (4.6) then becomes To write down the above double copy relations it is crucial that we have the following relation between the parity-odd and parity-even parts of the correlators : As we noted in the case of the double copy structure of T T O 3 in terms of JJO 3 the conformal Ward identity for correlators of the form J s J s O 3 does not have a non-homogeneous term. Hence the double copy relations that we arrived at here are purely for the homogeneous terms.

Double copy relation for general spin
One can easily extend the above analysis to correlators of the form J s J s O 3 in the spinor helicity variables, see [71]. The correlation functions are given by : The spinor helicity components of the odd part of the correlator can be computed using (4.7). One can then derive the following double copy relation for a general correlator of the kind J s J s O 3 that satisfies the homogeneous conformal Ward identity : We will now come to more complicated correlators such as JJJ and T T T whose conformal Ward identities have a non-homogeneous term and show more interesting double copy relations.

Double copy relation for JJJ and T T T
The double copy relation between JJJ and T T T is more subtle than those for correlators with a scalar operator insertion. Unlike T T O 3 or JJO 3 these correlators have a nonhomogeneous term as well and we will see that the double copy structures map homogeneous terms to homogeneous terms and non-homogeneous terms get mapped to non-homogeneous terms.

Homogeneous terms
The following double copy structure was noticed in [33] for the homogeneous term in the even part of T T T : T T T even,h = k 1 k 2 k 3 JJJ even,h JJJ even,h (4.10) From the explicit expressions for the correlators in (3.12), (3.13), and (3.18), we notice : We also have the remarkable relation that the parity-even part of the homogeneous term is given by the square of the odd part of the homogeneous term in JJJ : Combining these relations we obtain the following double copy relation for the complete homogeneous term of the T T T correlator :

Non-homogeneous terms
From (3.12) and (3.14) we know that JJJ even and T T T even also have non-trivial nonhomogeneous parts between which there exists the following double copy relation : This relation is independent of the double copy of the homogeneous part as the pre-factor is different. The non-homogeneous parts of T T T odd and JJJ odd are trivial as they are contact terms.

Double copy structure for higher spin correlators
We will now discuss the double copy structures in higher spin correlators. This is most easily done using the spinor-helicity variables. The parity-even part of the homogeneous part of J s J s J s is given by [71] : As noted in [71], the odd part of the above correlator is given by the same expression up to an overall factor of i : From this, we have the following double copy expression for the homogeneous part of the higher spin correlator J s J s J s : such that s + s = s.

Spin s current correlator as s copies of the spin one current correlator
In this section we note that we can write correlators of the form J s J s O 3 and J s J s J s as s copies of correlators of the spin-one current. Using the double copy relations in (4.10) recursively we notice that : Similarly using the double copy relations in (4.17) recursively we notice that :

CFT correlators from dS 4 Feynman diagrams and the flat space limit
In this section, we relate CFT correlators discussed in the previous section to the tree-level amplitude calculated using Feynman diagrams in dS 4 . We also relate CFT correlators to flat space scattering amplitudes.

Amplitudes
Here we will study 3-point flat-space scattering amplitudes in general parity-violating theories of gravitons, gluons and massless scalars in 4d. These are calculated straightforwardly from a Lagrangian whose cubic vertices will contribute to the 3-point amplitudes. We will use the notation M to denote the flat space amplitude without energy conservation. This is a useful quantity because E = k 1 + k 2 + k 3 = 0 in dS 4 and the dS 4 vertex is obtained from M by multiplying with an overall conformal time integral factor. This also matches the CFT correlators that we computed in Section 3. We also take flat space limit of M and resultant scattering amplitude will be denoted by A. More precisely We also express M in both 4d and 3d notations. See section 2 and the discussion below (2.1) for how to express 4d amplitude in terms of three dimensional notations. The 3d notation is particularly useful while comparing with the CFT correlators. We will first describe the gauge theory action, the gravity action and the gravity-gluon interactions that we consider.

Gauge theory action
The gauge theory action that we consider is : where S EM is the electromagnetic action and S A even and S A odd are gauge invariant paritypreserving and parity-violating actions respectively, given by : Depending on the background, g µν will be the four dimensional Minkowski or de Sitter metric.

Gravity action
The gravity action that we consider is : where S EH is the Einstein-Hilbert action and S g even and S g odd are parity-preserving and parity-violating actions respectively, given by : and W µνρσ is the Weyl tensor.

Gluon-gluon-scalar amplitudes
Let us first compute the contribution to the gluon-gluon-scalar amplitudes due to the interactions corresponding to the coupling α A 2 and β A 3 in (5.4) and (5.5). From the Lagrangian, we first compute the amplitude without energy conservation (M) and express it in both 4d and 3d notation (see (2.1)) we obtain To get actual flat space scattering amplitudes we have to impose energy conservation, i.e. E → 0. This gives :

Gluon-gluon-gluon amplitudes
Let us now compute the contribution to the gluon-gluon-gluon amplitudes due to interactions corresponding to couplings α A 1 and β A 2 in (5.4) and (5.5). The parity-even amplitude is given by : 14) The parity-odd amplitude is given by : The flat space scattering amplitudes is obtained by taking the E → 0 limit. This gives : We also have contributions to the 3-gluon amplitude from F 2 and F F . To obtain these we first calculate : The actual flat space scattering amplitudes are given by :

Gravity amplitudes
Working with a traceless, transverse perturbation around the flat metric, we obtain the following for the Weyl tensor to first order in perturbation : where k µ and z µ are defined in (2.1).

Graviton-graviton-scalar amplitude
Let us consider the graviton-graviton-scalar amplitude due to the interaction corresponding to the coupling β g 3 in (5.10). From the Lagrangian we have : The flat space amplitude is given by : Similarly, one can compute the graviton-graviton-scalar amplitude due to the interaction corresponding to the coupling α g 3 in (5.9) from M φW 2 : The flat space amplitude is obtained by taking the E → 0 limit :

Graviton-graviton-graviton amplitude
Here we calculate graviton-graviton-graviton amplitude from interactions corresponding to couplings α g 2 and β g 2 in (5.9) and (5.10). The parity-even amplitude is given by : The parity-odd contribution to the amplitude is given by : The flat space amplitude is obtained by taking the E → 0 limit : The EH term in (5.8) gives In the flat space limit this gives (5.28)

CFT correlators from dS 4
In this section we will compute tree level dS 4 cosmological correlators using the method of [56] and also including the relevant parity-odd 3-point vertices in the Lagrangian. The calculation with parity-even vertices was done in Appendix A of [33]. The idea, due to Maldacena and Pimentel [56] , that will be central to our analysis in this section is that certain cosmological correlators in de Sitter can be constructed directly from the corresponding flat-space amplitudes by dressing with conformal time integrals arising from using conformally covariant transformation properties of fields in dS 4 . These Lorentzian dS 4 correlators also compute boundary (Euclidean) CF T 3 correlators, thereby establishing a relationship between these three quantities. 5 We will work in (Lorentzian) dS 4 with the metric: For calculating dS 4 correlators perturbatively, we first look at the on-shell wave-functions for (linearised) free fields, which can be massless scalars, gauge or gravitational perturbations. These kind of fields will suffice to calculate the correlators we will be interested in. As is well known, for the scalar we have the Bunch Davies mode function φ ∼ (1−ikη) exp(ikη), the gauge field solution is a plane wave just like in flat space A µ ∼ z µ exp(ikη) and the linearised gravitational perturbation is given by γ µν ∼ z µ z ν (1 − ikη) exp(ikη). It was noted in [56], that this results in the Weyl tensors for the linearised gravity perturbations about dS and flat backgrounds being conformally related: whereas the gauge field strength is the same in both backgrounds. This means that the dS correlators of interest to us are the same as corresponding flat space amplitudes without energy condervation (M) upto conformal time integrals which are easily evaluated. In this section, we will calculate the contribution to 3-point functions from parity-violating interaction terms in the Lagrangian of the form F F φ, W W φ, F 2 F and W 2 W . We will take all parity-odd two-point functions to be vanishing as this is the case for the dS 4 actions considered here. We will find that our tree-level dS 4 computations will generate the different parts of the corresponding CF T 3 correlator (both parity-even and odd). Therefore, this perturbative approach provides a simple method of fixing the form of CFT correlators without taking recourse to solving conformal Ward identities.

JJO 3
The term in the action which contributes to the odd part of JJO 3 is given by : The tree-level dS 4 correlator corresponding to this interaction is given by 6 : Substituting for M φF F from (5.12) we see that this matches the expression for the homogeneous part of the correlator in (3.3). In the flat space limit we get : where A φF F is given in (5.13). The corresponding parity-even correlator is given by [20,29,33] : This matches the expression for the homogeneous part of the correlator in (3.2) if we use the expression for M φF 2 in (5.12). In the flat space limit the correlator takes the form : where A φF 2 is given in (5.13).

T T O 3
Let us now consider the parity-odd part of the correlator T T O 3 . The only contribution to T T O 3 odd comes from the term : The corresponding dS 4 correlator is given by : We can easily check that with the expression for M φW W in (5.20), this matches the homogeneous part of the correlator in (3.5). In the flat space limit we get : where A φW W is given in (5.21). The parity-even part of the corresponding correlator is given by [20,29,33] : which matches (3.4) if we use the expression for M φW 2 given in (5.22). In the flat space limit we get : where A φW 2 is given in (5.23).

JJJ
Let us now compute the odd part of JJJ . The contribution to JJJ odd comes from the following terms : The parts of the tree-level correlator corresponding to these interactions are given by : Combining the two, we get the total correlator to be : One can check using the explicit expression for M F 2 F in (5.15) that the first term in the above equation corresponds to the homogeneous term in (3.13). Similarly, using the explicit expression for M F F in (5.17) one can see that the second term in the above equation corresponds to the non-homogeneous term in (3.13). As noted earlier the term proportional to c odd JJ in (5.43) is a contact term. In the flat space limit we obtain : where A F F = 0 as shown in (5.18) and the expression for A F 2 F is as in (5.16). The even part of the correlator is given by [20,27,33] : From the explicit expression for M F 3 in (5.14) we see that the first term in the above equation corresponds to the homogeneous term in (3.12). Using the expression for M Y M in (5.17) one can see that the second term in the above equation corresponds to the nonhomogeneous term in (3.12). In the flat space limit we obtain : where the expression for A F 3 and A Y M are as in (5.16) and (5.18) respectively.

T T T
The contribution to T T T odd comes from the following term in the action : The tree-level dS 4 correlator corresponding to this term is given by : Using the result for M W 2 W given in (5.26), one can check that this matches the homogeneous part of the correlator in (3.5). This contribution was also calculated in [56]. In the flat space limit one obtains : where expression for A W 2 W is given in (5.27). The parity-even contribution to the correlator is given by [20,27,33] : which upon using M W 3 given in (5.24) and M EH given in (5.28) produces (3.14) upto contact term 7 .
In the flat space limit we obtain lim E→0 T T T even = c even T T T c 2 123 where A W 3 and A EG are given in (5.27) and (5.28).

Double copy structure of parity-violating amplitudes
In this section, we see how the double copy structure arises directly at the level of 3point scattering amplitude when parity-violating terms are taken into account. The 3-point amplitudes involved may be written down directly on symmetry grounds, or computed from the action given above. The double copy relations of amplitudes also follows from that for CFT correlators that we saw previously, but here we show that it can be demonstrated directly in 4d flat space . 7 This expression matches upto a number with the expression for the stress tensor correlator that appears in equation 4.12 of [33] once we identity See Appendix C.1 for details.

Scalar-graviton-graviton
From the momentum space expressions for M φF 2 and M φF F in (5.12) and the expression for M φW W and M φW 2 in (5.20) and in (5.22) we obtain : In the flat space limit when E → 0 one has :

Graviton-graviton-graviton
From the momentum space expressions for M F 3 and M F 2 F in (5.14) and (5.15) and the expression for M W 2 W in (5.26) one can verify that : In the E → 0 limit one has : We also have the following relation : The details of how one obtains this relation in momentum space are given in Appendix (C.1). Thus we notice that the following double copy structure extends beyond the E → 0 limit. We also notice using the momentum space expressions in (5.15) the following : Similarly one has the following double copy relation :

Discussion
In this paper we established various double copy relations for parity-violating CF T 3 momentum space 3-point correlators. The double copy structure is a very special property of CFT correlators in momentum space, the analogue of which does not exist in position space.
To understand this structure, we divided the momentum space CFT correlation function two parts, which we called homogeneous and non-homogeneous pieces. It was crucial for our analysis that the homogeneous part consist of two pieces of conformally invariant structures, namely one parity-even and one parity-odd structure whereas the non-homogeneous part has only one parity-even conformally invariant piece -all other contributions are contact terms. Squaring the homogeneous piece could in principle generate three structures. However, interestingly it turns out that squaring the parity-odd and even part produces exactly the same structure, whereas the cross-term which is generated by multiplying the parity-odd and parity-even part gives rise to the needed parity-odd structure.
This paper leaves various interesting directions to be followed upon. For example, for a general correlator of conserved currents of the form J s 1 J s 2 J s 3 , we need to understand its structure better to be able make any detailed statement about its double copy. For instance, if we want to understand the parity-odd contribution we need to understand analogue of triangle inequality [64]. We looked only at marginal scalars, but it is of interest to find out if the double copy structure shows up in J s J s O ∆ if O is a general scalar operator (that is, ∆ is arbitrary).
It is also of interest to see whether the double copy relations continue to hold for higher point functions [49]. Establishing double copy relations for 4-point functions is a significantly harder problem. In this case it would be interesting to see if the double copy structure is visible even in some tractable limit or if one can infer the existence of a double copy structure for specific conformal blocks or Polyakov blocks. In this paper we have focused mainly on conformally invariant structures but it would be interesting to see what kind of constraints one needs to impose on OPE coefficients to get double copy relations.
Another interesting direction to study is a momentum space analogue of the analysis of [64]. In [64] conformally invariant parity-even and parity-odd structures were identified in position space. These could then be appropriately composed to get conformally invariant parity-even and parity-odd correlators in position space. The double copy structure of the momentum space CFT correlators that we investigated in this paper already hints towards the existence of such structures in momentum space. See Appendix D for some details. We leave such questions for a future investigation.

Gluon-gluon-scalar amplitudes
In spinor-helicity variables, the non-zero components dS amplitude take the following form Also by complex conjugation ++ helicity results also exist. However here as well as below we don't write them explicity. In spinor-helicity variables flat space amplitudes the take the form where we have written down answer for negative helicity component. Although (A.1) and (A.2) might look similar, later is obtained from former in the E → 0 limit.

Graviton-graviton-scalar amplitudes
In spinor-helicity variables, the non-zero components of the dS 4 amplitude take the following form : In spinor-helicity variables flat space amplitudes the take the form

Gluon-gluon-gluon amplitudes
In spinor-helicity variables, the non-zero components of dS amplitudes are given by In the flat space limit we get

B T JJ
We calculate T JJ using gravity techniques for completeness. This result is not used in the main text.

Gluon graviton interaction
Let us consider interactions between gluons and gravitons which contribute to mixed CFT correlators of the spin-one conserved current J and the stress-tensor T . The interactions we consider are : are the parity-preserving and parity-violating actions, respectively. Tree level scattering amplitude for parity-even case is given by Let us now compute the odd part of T JJ . The contribution to T JJ odd comes from the following term in the action : This gives parity M W F F = [2(z 1 · k 2 )(z 2 · k 1 ) + 2(k 1 · k 2 )(z 1 · z 2 )] (z 1 k 1 z 3 k 3 ) = 2( z 1 · k 2 )( z 2 · k 1 ) + E(E − 2k 3 ) z 1 · z 2 (z 1 k 1 z 3 k 3 ) (B.5) The flat space amplitudes are obtained by taking E → 0 : The non-zero spinor-helicity components of the above ds amplitude are given by M −−+ In the spinor-helicity language the non-zero components of flat space amplitude are given by Now the parity-odd contribution to correlator T JJ is given by One can check that the correlation function given in (B.9) satisfies appropriate conformal Ward identity.
( k 2 · z 1 ) 2 ( k 3 · z 2 ) 2 + (k 2 The relation is given by (C.6) Using the same two degeneracies on (C.1) to remove the term proportional to E 2 , we obtain the following relation.
Therefore, we see that which says that the double copy holds beyond the flat space limit.

D Momentum space expression of higher spin correlators
In this section we give the momentum space expression for the parity-even and parity-odd parts of higher spin correlators J s J s O 3 and J s J s J s using relations in subsection 4.5 and